Product decomposition of periodic functions in quantum signal processing

Abstract

In some embodiments, one or more unitary-valued functions are generated by a classical computer generating using projectors with a predetermined number of significant bits. A quantum computing device is then configured to implement the one or more unitary-valued functions. In further embodiments, a quantum circuit description for implementing quantum signal processing that decomposes complex-valued periodic functions is generated by a classical computer, wherein the generating further includes representing approximate polynomials in a Fourier series with rational coefficients. A quantum computing device is then configured to implement a quantum circuit defined by the quantum circuit description.

What is claimed is: 
     
       1. A method, comprising:
 by a classical computer, generating one or more unitary-valued functions using projectors with a predetermined number of significant bits; and 
 configuring a quantum computing device to implement the one or more unitary-valued functions. 
 
     
     
       2. The method of  claim 1 , wherein the unitary-valued functions are iteratively decomposed. 
     
     
       3. A system, comprising:
 a quantum computing device; and 
 a classical computing device in communication with the quantum computing device and adapted to perform the method of  claim 1 . 
 
     
     
       4. A method, comprising:
 by a classical computer, generating one or more unitary-valued functions using projectors with a predetermined number of significant bits; and 
 configuring a quantum computing device to implement the one or more unitary-valued functions, wherein the unitary-valued functions are applied to implement a Hamiltonian simulation. 
 
     
     
       5. A method, comprising:
 by a classical computer, generating one or more unitary-valued functions using projectors with a predetermined number of significant bits; and 
 configuring a quantum computing device to implement the one or more unitary-valued functions, wherein the unitary-valued functions are applied to solve a linear equation. 
 
     
     
       6. A method, comprising:
 by a classical computer, generating a quantum circuit description for implementing quantum signal processing that decomposes complex-valued periodic functions, wherein the generating further includes representing approximate polynomials in a Fourier series with rational coefficients; and 
 configuring a quantum computing device to implement a quantum circuit defined by the quantum circuit description. 
 
     
     
       7. The method of  claim 6 , wherein the Fourier series with rational coefficients decreases numerical instability in the computation of decomposition. 
     
     
       8. The method of  claim 6 , wherein the quantum circuit description is configured to perform a Hamiltonian simulation. 
     
     
       9. The method of  claim 6 , wherein the quantum circuit description is configured to solve a linear equation. 
     
     
       10. A method, comprising:
 by a classical computer, generating a quantum circuit description for implementing quantum signal processing that decomposes complex-valued periodic functions, wherein the generating further includes truncating terms in approximation polynomials with coefficients determined by a preset accuracy parameter; and 
 configuring a quantum computing device to implement a quantum circuit defined by the quantum circuit description. 
 
     
     
       11. The method of  claim 10 , wherein the truncating decreases numerical instability in the computation of decomposition. 
     
     
       12. The method of  claim 10 , wherein the quantum circuit description is configured to perform a Hamiltonian simulation. 
     
     
       13. The method of  claim 10 , wherein the quantum circuit description is configured to solve a linear equation. 
     
     
       14. A method, comprising:
 by a classical computer, generating a quantum circuit description for implementing quantum signal processing that decomposes complex-valued periodic functions, wherein the generating further includes expanding factorized polynomial using discrete fast Fourier transforms; and 
 configuring a quantum computing device to implement a quantum circuit defined by the quantum circuit description. 
 
     
     
       15. The method of  claim 14 , wherein the expanding decreases numerical instability in the computation of decomposition. 
     
     
       16. The method of  claim 14 , wherein the quantum circuit description is configured to perform a Hamiltonian simulation. 
     
     
       17. The method of  claim 14 , wherein the quantum circuit description is configured solve a linear equation. 
     
     
       18. A method, comprising:
 by a classical computer, generating a quantum circuit description for implementing quantum signal processing that decomposes complex-valued periodic functions, wherein the generating further includes determining complementary polynomials by finding roots of Laurent polynomials to a preset accuracy; and 
 configuring a quantum computing device to implement a quantum circuit defined by the quantum circuit description. 
 
     
     
       19. The method of  claim 18 , wherein the quantum circuit description is configured to perform a Hamiltonian simulation. 
     
     
       20. The method of  claim 18 , wherein the quantum circuit description is configured solve a linear equation.